Triangle#Computing the area of a triangle

The exact calculation of the surface area of a triangular surface is one of the oldest problems of geometry. Already in ancient Egypt, it turned out, when, after the fall of the Nile flood the fertile farmland was to redistribute. Triangles and their area calculation still form an important basis of land surveying - using triangulation irregular surfaces can be determined. Even in modern fields of mathematics, the principle of triangular meshes used.

Your physical unit is the square meter ( m² ) on the spherical surface and the square degrees or steradians.

Area calculation on the planar triangle

Triangular faces can be calculated easily if you know the lengths of all three sides or the lengths of two sides and the included angle of them.

Where all three side lengths

If all three side lengths of a triangle are known, then can apply the theorem of Heron:

It is half the perimeter of the triangle.

It follows for half the circumference. This substituted into the above formula gives:

Given two side lengths and included angle

Two side lengths ( and the included angle ) of the triangle are known, then can the area of ​​the triangular area to determine a number of ways. The general formula for the area of ​​a triangle is

Here is the base side and the perpendicular thereto height of the triangle. The formula provides half the contents of a parallelogram, because every triangle can be supplemented his self to the corresponding parallelogram with a rotated copy. Whose area can be traced back by shear on the a rectangle. Another approach arises because a triangle can always be viewed as a special case of a trapezoid, wherein the second base side consists of only one point.

Although enables each side of the triangle to use as base side, the calculation of the corresponding amount is not elementary geometrically except in special cases possible. From trigonometry we deduce: . This results in:

Special cases

In right triangles, the height must not be charged extra. If the length of the other two sides is known, arises.

The height of an isosceles triangle with the legs always cuts the base in the middle and can therefore be calculated using the Pythagorean theorem and thus is

From the above-mentioned Formula also arises:

As a regular polygon, each equilateral triangle with an edge length of the level, this results in the area.

Because the angles are equal in an equilateral triangle all, it follows from the above Formula also:

Other cases

If the triangle is uniquely determined, possibly initially more angles or side lengths must be calculated until sufficient information is available for any of the above formulas.

Calculation with coordinates

In the plane

In the Euclidean plane with coordinate axes can be the area of a triangle with the points, and the trapezoidal formula derived. In the projection of the (possibly shifted in the first quadrant ) triangle on one of the axes, three trapezoids whose sum or difference is the triangular area arise. In this case, all the required parameters can be read at the elementary coordinates. For the area of ​​the triangle there is therefore:

This formula can be very clearly visualized with the aid of a determinant:

If you move the triangle so that is at zero, the result is due to the Laplace expansion theorem ( development by the first column ):

This second presentation in the form of determinants is also clear from the general volume formula for parallelepipeds, as a two-dimensional parallelepiped is a parallelogram with twice the triangle area. It is therefore important that the value of the determinant of a matrix whose columns are the vectors side of a triangle, twice the area provides this triangle. The same approach is obtained, when the triangle area is not the sum of trapezoidal surfaces, but as the sum of integrals over the three linear functions which define the three sides, perceives.

It is likewise possible to represent the three sides as curves in the plane, then the triangle is a piecewise smooth closed curve whose enclosed area can be calculated with the formula of Leibniz sector.

In three-dimensional space

In Euclidean space we obtain the area of ​​the triangle, which is spanned by and with the help of the cross product of the two vectors and. This provides a vector whose Euclidean norm is equal to the area of the parallelogram and clamped.

Area calculation of spherical triangles

Strictly speaking, no triangle on the Earth's surface is flat, because the earth is known, has an approximately spherical shape (see Earth's curvature ). For very large triangles ( about Cape Town - Rio de Janeiro - Tokyo ) one must therefore methods of spherical geometry to draw or the differential calculus: (or sphär trigonometry. )

By the theorem of Legendre a small spherical triangle has almost the same area as a flat triangle with three equal sides. These so-called Verebnung is more accurate, the smaller the triangles. It follows an iterative method and calculating the size of a spherical triangle: you halve repeated the geodesic lines that form the boundary of the triangle, and calculate the resulting from the smaller triangles area totals. The limit of this process exists, and the area of ​​the spherical triangle.

Two direct paths lead of course faster to your destination either by suitable formulas of spherical trigonometry or via the spherical excess ( the excess of the sum of the angles about 180 °). For a spherical triangle with internal angles, which lies on a sphere with radius, while the following formula applies:

The excess is directly proportional to the delta area, which is accurate enough on the Erdellipsoid for the practice of geodesy. The replacement of ball triangles by their planar equivalents, however imprecise starting at about 10 km.